This invention relates generally to seismic data processing and more particularly to a method for compensating for the effects of irregular spatial data sampling of seismic wavefields and of irregular illumination of subterranean seismic reflectors by seismic wavefields.
In seismic exploration, an acoustic wavefield (a shot) is generated by an acoustic source. The wavefield propagates through the earth from a source location. The wavefield is reflected from earth layers beneath the surface whence it returns to the surface. A plurality of seismic detectors are distributed on or near the surface of the earth, remotely from the source location, along lines of survey or in large areal patches. Preferably, the detectors, which constitute discrete wavefield sampling stations, are uniformly distributed spatially so that the wavefield can be uniformly sampled both areally and temporally. The sampled data are quantized and archivally recorded for further processing.
Fundamental to most of the methods for imaging of subsurface reflections is the movement, or xe2x80x9cmigrationxe2x80x9d of the data from the location where they are recorded to the location where the subsurface reflection originates. Kirchoff migration is currently the most commonly used migration algorithm for 3-D prestack depth migration. The Kirchoff integral is based upon a high frequency approximation of the wavefield and is based upon associated traveltime computations. For this reason, it is not able to adequately handle the most complex situations. In any case, the integral is, in prior art applications, evaluated as though the wavefield is regularly sampled. In reality, this is far from being the case for a number of reasons. First, the sampling of the data at the surface is not uniform because of obstructions such as buildings, roads or other culture. In the case of marine exploration, the distribution of receiver locations may be irregular because of errors in the assumed detector locations due to cable drift because of currents and wind or due to the presence of drilling and production platforms. The seismic data are often not only locally under-sampled, they also may be locally excessively densely sampled. A second problem with the evaluation of the Kirchoff integral is that where, as is almost always the case, the velocity of propagation in the subsurface is non-uniform, the illumination of the reflectors by the seismic wavefield is also non-uniform, whereas when the Kirchoff integral is evaluated, the illumination of the reflectors is usually assumed to be uniform.
Well-known seismic processing methods such as stacking, multi-channel filtering, dip moveout correction (DMO), prestack migration, velocity analyses, anisotropy studies, migration and wavefield extrapolation, all assume that the data are uniformly sampled. However, as noted above, the data gathered may be irregularly sampled. That irregularity may be due to obstructions as earlier explained or to missing shots or to inoperative detectors or receivers. When such irregular or inadequately spatially sampled data are not corrected, unwanted computational artifacts may result that are superimposed upon the processed output data.
Wave-equation processing routines such as DMO and movement of data samples, such as by repositioning because of dip migration, can be affected by irregular spatial sampling. In fact, the algorithm meant to improve the output data actually degrades that data because spatial under-sampling or over-sampling leaves remnants of the data-processing operators in the output data. Gross under-sampling, of course may result in dip aliasing. Noise, which is offset-dependent, is only partly canceled out in the stacking operation in the presence of locally-sparse spatial sampling.
In this disclosure, the term xe2x80x9coperatorxe2x80x9d will be used frequently. The term is defined to mean a specific thing involved in a data-processing operation. Thus a DMO operator is a specific expression involved in applying a correction to normal moveout for dip. An operator may be expressed as a symbol indicating an operation to be performed and itself may be the subject of mathematical manipulation.
Various authorities have addressed the problem of locally-sparse spatial wavefield sampling. In a paper entitled xe2x80x9cWave-equation Trace Interpolationxe2x80x9d, (Geophysics, vol. 52, no. Jul. 7, 1987, pp 973-984) J. Ronen discloses that a processing sequence in which one treats missing data as zero data and performs partial migration before stacking is equivalent to application of the transpose of the stacking operator that actually needs to be inverted. Ronen states that the inverse of the operator cannot be uniquely determined but it can be estimated using spatial spectral balancing in a conjugate-gradient iterative scheme. The first iteration is simply prestack partial migration. Where spatial aliasing is present, several additional iterations are needed.
R. G. Williams et al, in a paper delivered at the 51st annual meeting of the EAEG, May 1989, entitled xe2x80x9cModel-constrained Anti-alias Filtering for Improved DMOxe2x80x9d, apply an anti-alias filter to the data so that the azimuthal under-sampling of the DMO operator can be reduced by use of a dip model for the survey to limit the operator aperture in an azimuth-dependent manner.
In a paper published as expanded abstract no. 1144, at the 59th International Meeting of the SEG, 1989, entitled xe2x80x9cEffect of Irregular Sampling on Prestack DMOxe2x80x9d, J. Black et al. explain that irregular midpoint and azimuthal distributions of the seismic data will cause artifacts in the DMO output if those irregular distributions are ignored. That is said to be especially a problem for pre-stack DMO with land data-collection geometries. Using the actual data-collection geometry, the DMO response can be computed for flat events and can be used to design long wavelength corrections and selective edits that minimize the impact of DMO-induced artifacts. This procedure can be used with any DMO implementation but it is especially relevant for 3-D DMO using the Kirchoff method.
U.S. Pat. No. 5,206,837 issued to Beasley et al. disclose a method based on the decomposition of a seismic-wavefield gather into its constituent dip and source-detector offset components, that accounts for the effect of irregular spatial data sampling by applying an inverse sampling operator to the gather. The method can be applied to any 2-D or 3-D DMO algorithm in a multi-channel wavefield-gathering operation.
In Beasley et al. a wavefield is propagated from a source location. The wavefield is spatially sampled at a plurality of discrete data-sampling stations that are distributed over an area that is remote from the source location. The wavefield samples are combined into a raw gather to which a selected processing operator is applied in a processing step to provide a processed gather. An inverse sampling operator is then applied to the processed gather to provide an equalized wavefield gather. The inverse sampling operator includes the steps of decomposing the processed gather into constituent dip components. The movement and scaling of selected attributes of the components, resulting from the processing step, are analyzed. The movement and scaling of those components are corrected for undesired processing artifacts due to locally, relatively sparse, or relatively dense spatial sampling. In an aspect of this invention, the processing operator may be a form of DMO. While the equalization method in Beasley may be applied to any multichannel data processing regime such as anisotropy studies, filtering operators, migration and wavefield extrapolation, there are no explicit examples of the use of the equalization method to compensate for irregular sampling of the wavefield at the reflecting interfaces due to velocity irregularities.
The present invention shows how the equalization method disclosed in Beasley may be used to correct for irregular sampling of the seismic wavefield due to velocity irregularities, thereby giving an improved Kirchoff migration wherein amplitudes are better preserved.